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A077417 Chebyshev T-sequence with Diophantine property.

%I A077417 #63 Aug 30 2025 12:35:11
%S A077417 1,11,131,1561,18601,221651,2641211,31472881,375033361,4468927451,
%T A077417 53252096051,634556225161,7561422605881,90102515045411,
%U A077417 1073668757939051,12793922580223201,152453402204739361,1816646903876649131,21647309444315050211
%N A077417 Chebyshev T-sequence with Diophantine property.
%C A077417 7*a(n)^2 - 5*b(n)^2 = 2 with companion sequence b(n) = A077416(n), n>=0.
%C A077417 a(n) = L(n,12), where L is defined as in A108299; see also A077416 for L(n,-12). - _Reinhard Zumkeller_, Jun 01 2005
%C A077417 [a(n), A004191(n)] = the 2 X 2 matrix [1,10; 1,11]^(n+1) * [1,0]. - _Gary W. Adamson_, Mar 19 2008
%C A077417 Hankel transform of A174227. - _Paul Barry_, Mar 12 2010
%C A077417 Alternate denominators of the continued fraction convergents to sqrt(35), see A041059. - _James R. Buddenhagen_, May 20 2010
%C A077417 For positive n, a(n) equals the permanent of the (2n)X(2n) tridiagonal matrix with sqrt(10)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - _John M. Campbell_, Jul 08 2011
%C A077417 Positive values of x (or y) satisfying x^2 - 12xy + y^2 + 10 = 0. - _Colin Barker_, Feb 09 2014
%C A077417 a(n) = a(-1-n) for all n in Z. - _Michael Somos_, Jun 29 2019
%H A077417 Vincenzo Librandi, <a href="/A077417/b077417.txt">Table of n, a(n) for n = 0..200</a>
%H A077417 Alex Fink, Richard K. Guy, and Mark Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
%H A077417 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A077417 J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv:1403.5962 [math.CO], 2014.
%H A077417 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-1).
%H A077417 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A077417 a(n) = 12*a(n-1) - a(n-2), a(-1)=1, a(0)=1.
%F A077417 a(n) = S(n, 12) - S(n-1, 12) = T(2*n+1, sqrt(14)/2)/(sqrt(14)/2) with S(n, x) := U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120. S(-1, x)=0, S(n, 12)=A004191(n).
%F A077417 G.f.: (1-x)/(1-12*x+x^2).
%F A077417 a(n) = (ap^(2*n+1) + am^(2*n+1))/sqrt(14) with ap := (sqrt(7)+sqrt(5))/sqrt(2) and am := (sqrt(7)-sqrt(5))/sqrt(2).
%F A077417 a(n) = sqrt((5*A077416(n)^2 + 2)/7).
%F A077417 a(n)*a(n+3) = 120 + a(n+1)*a(n+2). - _Ralf Stephan_, May 29 2004
%F A077417 E.g.f.: exp(6*x)*(7*cosh(sqrt(35)*x) + sqrt(35)*sinh(sqrt(35)*x))/7. - _Stefano Spezia_, Aug 29 2025
%e A077417 G.f. = 1 + 11*x + 131*x^2 + 1561*x^3 + 18601*x^4 221651*x^5 + 2641211*x^6 + ...
%t A077417 CoefficientList[Series[(1 - x)/(1 - 12 x + x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Feb 10 2014 *)
%t A077417 LinearRecurrence[{12,-1},{1,11},30] (* _Harvey P. Dale_, Apr 09 2015 *)
%t A077417 a[ n_] := With[{x = Sqrt[7/2]}, ChebyshevT[2 n + 1, x]/x] // Expand; (* _Michael Somos_, Jun 29 2019 *)
%o A077417 (Magma) I:=[1,11]; [n le 2 select I[n] else 12*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Feb 10 2014
%o A077417 (PARI) my(x='x+O('x^30)); Vec((1-x)/(1-12*x+x^2)) \\ _G. C. Greubel_, Jan 18 2018
%o A077417 (PARI) {a(n) = my(x = quadgen(56)/2); simplify(polchebyshev(2*n + 1, 1, x)/x)}; /* _Michael Somos_, Jun 29 2019 */
%Y A077417 Cf. A072256(n) with companion A054320(n-1), n>=1.
%Y A077417 Row 12 of array A094954.
%Y A077417 Cf. A004191.
%Y A077417 Cf. A041059. [_James R. Buddenhagen_, May 20 2010]
%Y A077417 Cf. similar sequences listed in A238379.
%K A077417 nonn,easy,changed
%O A077417 0,2
%A A077417 _Wolfdieter Lang_, Nov 29 2002
%E A077417 More terms from _Vincenzo Librandi_, Feb 10 2014